r\A  \  vi-n  Lo  ^  IY  9  l 


THE 


ASTROPHYSICAL  JOURNAL 


AN  INTERNATIONAL  REVIEW  OF  SPECTROSCOPY 
AND  ASTRONOMICAL.  PHYSICS 


JANUARY 


NUMBER  i 


VOLUME  VII 


THE  SYSTEM  OF  £  LYRAE. 


By  G.  W.  Myers. 


Of  the  various  theories  which  have  been  proposed  to 
explain  the  light  changes  of  variables,  three  have  been  most 
widely  accepted  : 

First.  A  body  whose  surface  possesses  different  degrees  of 
brightness^  different  places,  rotates  upon  an  axis,  bringing  the 
differently  illuminated  portions  into  view  at  regular  intervals. 

Second.  A  secondary  meteoric  swarm  circles  about  a  primary 
swarm,  so  as  to  pass  at  regular  intervals  between  the  observer 
-  and  the  central  swarm.  The  outlying  meteors  .of  the  two 
"  swarms,  colliding  with  each  other  at  the  times  of  periastron 
-"passage,  produce  also  a  periodic  increase  of  brightness.  A  suit- 
^able  combination  of  these  two  causes  of  light  variation  will 
explain  a  large  variety  of  the  light  fluctuations  observed  in  varia- 


Third.  The  so-called  satellite  theory,  in  which  two  bodies 
whose  luminous  intensities  may  be  either  equal  or  unequal, 
revolve  about  each  other  in  orbits  whose  planes  pass  near  the 
solar  system,  and,  bv  the  mutual  eclipses  of  the  bodies,  light 
changes  of  such  character  are  produced  as  to  explain  some  sort 
of  variability. 

1  Read  at  the  conferences  held  in  connection  with  the  dedication  of  the  Yerkes 
Observatory,  Oct.  20,  1897. 


P 


lba<;5 


2 


G.  W.  MYERS 


The  arbitrariness  involved  in  the  fundamental  assumptions 
of  theories  one  and  two,  appears  to  me  to  be  an  element  of 
weakness  of  so  serious  a  nature  as  to  compel  them  to  yield 
to  the  third  theory  in  all  cases  where  the  latter  is  applicable. 
By  proper  assumptions,  indeed,  regarding  the  extent,  mode 
of  distribution,  and  periodicity  of  the  spots,  or  respecting  the 
periodicity  and  distribution  of  the  meteors  within  the  swarms, 
as  also  the  relative  motions  of  the  swarms  themselves,  any  sort 
of  light  change  is  capable  of  explanation  by  the  first  two  theo¬ 
ries  ;  but  as  has  been  pointed  out,  their  well-nigh  universal  appli¬ 
cability  is  in  itself  an  element  of  weakness,  inasmuch  as  the 
theories  in  themselves  are  in  no  essential  particular  an  advance 
on  their  underlying  hypotheses. 

It  is  my  purpose  to  give,  in  a  few  words,  an  account  of  a 
recent  attempt  to  represent  the  light  changes  of  ft  Lyrae,  on  the 
basis  of  the  so-called  satellite  theory.  Argelander’s  third  curve 
of  this  star  was  based  upon  I  500  careful  photometric  estimates, 
extending  over  a  period  of  nineteen  years.  An  earlier  curve  by 
Oudemans  and  a  later  by  Schoenfeld,  show  no  discrepancies 
from  this  curve  of  a  magnitude  great  enough  to  affect  the  dis¬ 
cussion  appreciably,  and,  consequently,  Argelander’s  third  curve, 
published  in  a  pamphlet  entitled,  “  De  Stella  /3  Lyrae  Variabili 
Commentatio  Altera”  in  1859,  was  taken  as  the  basis  for  the 
discussion.  This  well-known  curve  is  represented  in  the  upper 
portion  of  the  accompanying  figure. 

INTRODUCTORY. 

The  variability  of  this  star  was  discovered  by  Goodericke  in 
1784,  but  aside  from  the  recognition  of  the  fact  of  its  variability 
and  the  confirmation  of  the  existence  of  two  unequal  minima  by 
this  same  scientist,  nothing  further  was  done  in  its  study  until 
Argelander  laid  the  foundation  for  a  thoroughly  scientific  and, 
at  the  same  time,  an  extremely  practical  method  of  studying 
the  light  fluctuations  of  variables.  Applying  his  own  method 
to  Lyrae,  Argelander  showed  the  star  to  have  two  unequal 
minima,  separated  by  two  practically  equal  maxima,  the  entire 


SYSTEM  OF  /3  LYRAE 


3 


cycle  of  changes  being  completed  in  about  I2d  22h  (  =  I2d.gi). 
At  the  time  of  the  I  Maximum,  he  found  the  brightness  of  the 
star  to  rise  to  the  3.4  magnitude,  and  to  fall  after  about  three 


days  to  a  secondary  minimum  of  the  3.9  magnitude,  to  rise  again 
after  three  days  to  the  former  brightness  of  the  3.4  magnitude 
and  then,  after  nearly  the  same  interval,  to  fall  to  a  primary 
minimum  of  the  4.5  magnitude.  Or,  in  other  words,  the  bright¬ 
ness  at  the  maximum  corresponds  to  12.33  of  Argelander’s 


4 


G.  W.  MYERS 


grades  (0.127  mag.);  at  the  primary  minimum,  to  3.34  grades 
and  at  the  secondary  minimum,  to  8.53  grades. 

Assuming  the  light  changes  to  be  due  to  the  mutual  eclipses 
of  two  revolving  bodies  of  unequal  brightness,  we  should  have 
the  maxima  occurring  when  the  components  stand  beside  each 
other,  without  either  being  eclipsed  by  the  other,  and  conse¬ 
quently  both  disks  shining  full-phase.  At  the  primary  mini¬ 
mum,  the  darker  component  lies  in  front  of  the  brighter  and 
cuts  off  a  portion  of  the  latter’s  light,  while  at  the  secondary 
minimum  the  bright  companion  passes  before  the  darker  and 
obscures  a  part  of  its  inferior  brilliance.  The  relative  positions 
of  the  components  at  the  chief  epochs  are  shown  in  the  draw- 
ing  (Fig.  2). 


Since  the  brightness  of  the  “system  ”  is  reduced  at  Minimum 
I  (primary  minimum)  by  66  per  cent,  of  its  greatest  brightness 
and  at  Minimum  II  (secondary  minimum)  by  only  36  per  cent., 
it  is  evident  (i),that  the  disks  must  be  assumed  unequally 
bright  or  (2),  that  the  orbital  eccentricity  must  be  assumed 
great  or  finally,  that  both  these  circumstances  concur. 


SYSTEM  OF  ft  LYRAE 


5 


ECCENTRICITY. 

An  approximate  idea  of  the  magnitude  of  the  eccentricity 
may  be  obtained  in  the  following  two  ways  : 

First.  If  we  assume  the  motions  of  the  components  to  be 
in  conformity  to  Kepler’s  laws  we  have,  for  the  relation  of  the 
true  and  mean  anomalies,  the  well-known  equation  : 

v  —  M- f-  2 c.  sin  M- f-  -f  e 2  sin  2 M-f-  .... 

The  lengths  of  the  chief  intervals  of  light  change,  from  Arge- 
lander’s  curve  are  : 


Min.  I  — Max.  I  —3.125  days. 
Max.  I  — Min.  II  — 3.250  i( 


Min.  II  - —  Max.  II  —  3-167  “ 

Max.  II  —  Min.  I  =  3.368  “ 


The  approximate  equality  of  these  intervals  points  unmis¬ 
takably  to  a  small  orbital  eccentricity.  Assuming  now,  the 
eccentricity  to  be  so  small  as  to  render  its  higher  powers  negligi¬ 
ble,  we  may  shorten  the  above  equation  into  : 

(a)  v  =  M - \-  2e  sin  M. 

Designating  the  respective  values  of  v  and  M at  Min.  I,  Max. 
I,  Min.  II  and  Max.  II  by  vlf  Ml  ;  v2,  M2  ;  v3,  M3  and  v4,  M4 , 
and  substituting  in  (#),  there  result  the  following  four  equations  : 


(1)  vx  =MX  -f  2<f.  sin  M1 

(2)  v2  —  M2  q-  2e.  sin  M2 


(3)  =  M3  -(-  2c .  sin  M3 

(4)  v 4  =  M4  +  2e.  sin  A/4 


6 


G.  IV.  MYERS 


Subtracting  (i)  from  (2),  (3),  and  (4)  in  succession,  and 

. 


noting 

that  v2- 

~v \  — vi  = 

=  7r ;  and  — - 

obtain 

(s) 

7 r 

2 

-  2  mx  -)-  4  c.  cos  (Mx 

-f-  vi x)  sin  mx 

(6) 

7 r 

-  2mtl-\-  /[e.cos  (Mx- }- 

m 2 )  sin  ;;/2 

(7) 

7 r 

3  2 

-  2  m  %  -f-  4  e-  cos  (Mx 

-f-  ^3)  sin 

where 

2  VI  j 

=  M^—MX  =  3.125/x, 

=  37°  8 '  .4 

2  VI 

=  MS  —  MX  6.375/A 

=  1 77 °  46 '.2 

2  W8 

=  M^  —  M\  =  9.542/A 

266°  4'.  8 

and 

fX 

=  2tt  T5,  7J 

12.91  days. 

we 


We  shall  then  have  but  two  unknowns  in  (5),  (6)  and  (7),  viz., 
e.  and  Mx,  and  any  pair  will  suffice  to  determine  these  magni¬ 
tudes. 

From  (5)  and  (6)  we  find, 

(M  j^i  =  -3°°28' 

[  >  |  0.0186, 

and  these  values  substituted  in  (1),  give 
vi=—  3J°  33'- 


This  satisfies  the  intervals  Max.  I — Min.  I,  and  Min.  II — Min.  I, 
and  requires  Min.  I  to  lie  310. 5  before  periastron. 

From  (5)  and  (7),  there  result,  similarly: 

(  Mx  —  209 ° 3 2 ' 4 
(y)  -j  e  =  0.0196 

l  Vx  =  209°24'  . 

Computing  v  x  from  Argelander’s  curve,  it  will  be  found  that 
a  rigorous  satisfaction  of  the  intervals  Min.  I  to  Max.  I  and  Min.  1 
to  Min.  II,  requires  a  backward  displacement  of  Argelander’s 
Max.  II  by  about  4  hours,  while  the  intervals  Min.  I — Max.  I 
and  Min.  I — Max.  II,  used  in  getting  (*:),  necessitate  a  backward 
displacement  of  Max.  I  by  about  4  hours.  Since  now,  so  small 
a  shift  in  these  two  chief  epochs,  corresponds  to  so  large  a 


SYSTEM  OF  LYRAE 


7 


change  (nearly  i8o°)  in  the  position  of  periastron,  the  eccen¬ 
tricity  of  the  orbit  cannot  be  large.  The  mean  of  the  two  nearly 
equal  values  of  the  eccentricity  found  above,  is  o. 0191  ±0.0033. 
This  value  is  small  enough  to  justify  neglecting  its  second  and 
higher  powers  as  was  done  above  and  thereby  vindicates  the 
method  of  treatment. 

Second.  It  will  develop  later  that  the  hypothesis  of  a  flatten¬ 
ing  of  one  or  both  bodies  must  be  made.  Assuming  the  bodies 
to  be  deformed  by  reciprocal  tidal  influence,  or  by  whatever 
cause,  into  similar  ellipsoids  of  revolution  —  a  permissible  assump¬ 
tion,  since  such  forms  are  figures  of  equilibrium- — and  denoting 
the  ratio  of  the  major  and  minor  axes  by  “  q ,”  so  soon  as  an 
approximate  value  of  q  is  known,  a  superior  limit  of  the  ratio  of 
distance  of  centers,  to  the  larger  semi-major  axis  may  be  derived. 
Thus,  assuming  the  bodies  to  be  spheres,  with  radii  equal  to 
the  respective  semi-minor  axes  of  the  ellipsoids,  a  light  curve 
due  to  two  revolving  spheres  may  be  computed.  A  lower  limit 
for  the  duration  of  the  eclipse  at  Min.  I,  for  example,  may  be 
read  off  from  this  curve.  A  little  reflection  will  show  that  the 
larger  q  be  taken,  the  shorter  the  duration  of  the  eclipse  will  be. 
Taking  then,  a  value  of  q  greater  than  that  tound  later  to  be 
the  approximate  value  and  assuming  that  the  eclipse  has  not 
begun  until  the  light  curve  has  fallen  considerably,  a  value  for 
the  eclipse-duration  will  be  obtained  which  is,  at  all  events, 
small  enough,  perhaps  much  too  small,  q  is  later  found  to  be 
1.2,  and  assuming  it  to  be  1.3,  I  find  for  inferior  limit  of  eclipse- 
duration,  3  days  and  4  hours,  which  must  be  at  all  events,  small 
enough.  But  the  smaller  this  inferior  limit,  the  larger  must  be 
the  ratio  of  distance  of  centers  to  the  larger  semi-major  axis. 
Using  the  above  value  3d  4h  for  eclipse-duration,  a  superior  limit 
for  this  ratio  may  then  be  obtained,  which  will  at  any  rate,  be 
large  enough.  Taking  now,  the  larger  radius  as  unity  and 
denoting  the  smaller  by  *,  the  radius  vector  of  the  true  orbit 
by  r  ( for  this,  e  may  be  put  -  o) ,  one  half  the  distance  between 
the  nearest  points  of  the  satellite  at  the  beginning  and  end  of 
the  eclipse,  by  we  see  from  the  figure  that, 


8 


G.  IV.  MYERS 


CPC'  ^3. 167X27°.  887  =  88°  20 ' 
and  hence  CPD  >44°- 10'. 

Also  r  V  (jv  T  k)  c  s  c  440  io' 

1(X  +  K)  I.438. 

But  since  x^i ,  I,  we  haver<2.87  times  the  smaller 
semi-axis  of  the  larger  ellipsoid  or  =  2.4  times  its  larger  semi¬ 
axis.  The  distance  between  centers  then,  being  so  small  com¬ 
pared  with  the  dimensions  of  the  larger  body,  the  eccentricity 
could  not  be  large  or  the  masses  would  necessarily  interpene¬ 
trate  during  revolution. 

These  considerations  are  sufficient  to  warrant  the  assump¬ 
tion  of  a  small  orbital  eccentricity  and  to  justify  the  hypothesis, 
that  a  first  approximation  to  the  orbit  may  be  obtained  by  mak¬ 
ing  e  =  0. 

FLATTENING. 

The  necessity  of  the  assumption  either  that  the  disks  are 
flattened,  or  that  the  bodies  are  not  yet  separated,  is  apparent 
from  the  fact  that  the  brightness  at  the  maxima,  does  not  remain 
constant  for  any  considerable  length  of  time.  It  has  been 
assumed  in  what  follows  that  : 

( 1 )  The  two  bodies  are  distinct  and  separate. 

(2)  Both  are  deformed  into  ellipsoids  of  revolution. 

(3)  The  periods  of  rotation  and  of  revolution  are  equal  to 
I2d.9I. 

(Possible  librations  being  disregarded.) 


SYSTEM  OF  /3  LYRAE 


9 


Taking  the  origin  of  coordinates  at  the  center  of  the  larger 
ellipsoid  and  assuming  the  brightness  of  the  disks  uniform,  we 
may  readily  find  from  the  principles  of  analytical  geometry  of 
space,  the  equation  of  the  light  curve,  which  may  be  used  while 
the  bodies  stand  wholly  off  each  other.  It  was  found  to  be 


X 


i  +  A  j  |  /sin2  <£  +  (R/R'y  cos2  <j> 

+  —  =  l 

I  sin2  <£  +  (B / B'Y  cos2  (j>  \ 

Where  L  denotes  the  ratio  of  the  combined  luminosities  of 
both  simultaneous  elliptical  disks  to  the  combined  brightness 
when  the  areas  of  these  disks  are  greatest,  i.  e.y  at  the  instant 
of  the  maxima  : 

A.  is  the  ratio  of  the  intensities  of  the  disks, 

k  is  the  ratio  of  the  corresponding  semi-axes  of  the  two  ellipsoids. 

B' ,  R'  and  B,  R  denote  respectively  the  semi-major  and  semi¬ 
minor  axes  of  the  smaller  and  larger  ellipsoids,  6bcisthe  sight  line  and 
ZOy  the  tangent  plane  to  the  celestial  sphere  at  the  system. 


o 


G.  W.  MYERS 


cf>  is  the  longitude  of  the  secondary  in  its  orbit. 

As  a  first  approximation  q  was  put  equal  to  R' / R  =  B' /  B. 

From  the  unsymmetrical  character  and  small  magnitude  of  the 
final  residuals  furnished  by  a  comparison  of  Argelander’s  curve 
with  the  computed  curve,  it  appears  later,  that  nothing  can  be 
gained  by  an  attempt  to  improve  this  hypothesis  by  ascribing 
different  degrees  of  flattening  to  the  two  ellipsoids,  so  long  as 
both  bodies  be  regarded  symmetrical  and  their  disks  uniformly 
bright.  On  this  hypothesis  the  above  equation  becomes 

i  i 

Z=  7  0  _  . ,  where  cf>  —  - — -  t  U  being  in  hours). 

V sin2  cf>  +  i  / q 2  cos2  12  P  v  6  ’ 

An  approximate  value  of  q  was  found  by  computing  light 
curves  for  various  values  of  q ,  viz.,  i.i;  1.2;  1.25;  and  1.3, 
through  a  number  of  symmetrically  chosen  points  before  and 
after  the  maxima,  and  deducting  the  computed  values  of  the 
ordinates  from  those  of  Argelander’s  curve.  The  correct  value 
of  q  should,  of  course,  give  a  curve  whose  ordinates,  deducted 
from  the  corresponding  ordinates  of  Argelander’s  curve,  would 
be  those  of  a  curve  due  to  two  revolving  spheres,  and  for  a  time 
on  either  side  of  the  maxima,  i.e .,  while  the  two  components  are 
wholly  uncovered,  such  a  curve  must  run  horizontally.  The 
value  1.2  of  q  gave  the  following  nearly  equal  ordinates  : 

10.33;  10.55;  10.76;  10.78;  10.78;  10.79;  10.78;  10.78;  10.76; 
10.55  ;  and  IO-33- 

This  value  of  q  was  then  assumed  as  a  first  approximation. 

An  equation  of  a  light  curve  applicable  during  the  eclipses 
was  then  computed  by  the  process  suggested  in  the  drawing, 
and  for  Min.  I,  the  necessary  equations  were  found  to  be  : 

J‘  '  -  .77“+  k-  a,  j  *  +  (»-*')- P‘ ’ si®  *  [ 

,  p  sin  (cb±<b')  ,  ..  o  .  „  .  o 

pr  .  7  ;  according  as  <f>  <  90  or  <f>  >>  90  , 


SYSTEM  OF  /3  LYRAE 


I  I 


Putting  M,=  tt(i  +  k2  X)  (i — //). 

Then  j-  k2  <£"  — p'  sin  <j>  or  J/7—  -|-  *2  —  #c  sin(<£"  +  <£). 


The  work  of  computation  was  somewhat  shortened  by  intro¬ 
ducing  an  additional  auxiliary  Hh  when  <£">^  and  by  using  the 

foregoing  equations  only  when  (/>"<-.  Hj  was  defined  by 
Hj=  K2  7T—  7T  (i  -f  K2  A.)  (i - //), 

Whereupon 

Hj—k 1  <£'-*—<£ — p'  sin  <£,  or  H1  =  k 2  <£' —  </>  -j-  k  sin  (<j>' —  <£) 
and  similarly,  the  equations  for  Min.  II  were  found  to  be  : 

^"'=I_,r(i+,c*  A)  {  <A  +  k'2  (*-  +  ')-p'sta  *  ( 

1  1  ■  K 2  A.  i  / 

MfI=ir — — - j  1 — Jn  ,  or  —  -f-  k2  <f)lf — p'  sin  <f> 

Hu  =  k2  tt  —  7T-1  — -  (i — ///'),  or  K2  <£" —  <£-fV  sin  <£. 

A 

The  remaining  formulae  hold  without  modification  for  both 
minima.  Jj '  and  JIfr  are  the  ratios  of  the  combined  instantaneous 
brightness  to  the  combined  brightness  due  to  the  sum  of  the 
two  instantaneous  disks  shining  full-phase  during  the  eclipses  at 
Min.  I  and  Min.  II  respectively. 


G,  W.  MYERS 


I  2 


Jr  and  Jir  are  the  ratios  of  the  same  brightness  to  that  due 
to  the  sum  of  the  full  disks,  when  these  disks  are  of  maximum 
area. 

The  relation  between  J/  and  /7  on  the  one  hand,  and  J n' 
and  JIf  on  the  other,  were  then  derived  and  found  to  be 

Ji=fJ/  and  Jn=fJn' ,  where  f=  R/  /R  =  B/  /B 

i 

j  '"sin2  /?-f-  ^2  cos2  /?’ 

and  /3  cf)  —  -  denotes  the  longitude  in  the  orbit,  counted  from 

a  point  90°  ahead  of  the  origin  of  (f>.  (For  undefined  symbols 
see  Fig.  6.) 

Designating  by  /3-|-  ^  ,  the  true  anomaly  in  the  real  orbit,  and 

by  a  that  in  the  apparent  orbit,  both  counted  from  the  node,  and 
calling  p  and  r  the  radii  vectores  in  the  apparent  and  true  orbits 
respectively,  we  have 

r.  sin  (3  =  p  cos  a  and  tan  a  =  cos  i  cot  /3, 
wherefore  p‘2=r2.  sin2  fi-\-r2.  cos2  i.  cos  (3 . 


Differentiating  the  formulae  for  Min.  I  and  reducing,  we  find 
8Mf  SB, 


8  <f> 


and  8  </> 


2  k  t g  (f)"  sin  ((f)”  -f-  (f))  ~~  ^  2  Ktg  <f)'  sin  (<f) '  —  <£)’ 

and  when  tcy>  1,  as  will  be  found  to  be  the  case  later,  the  pre¬ 
ceding  equations  become 

Mj=  (f>  -j-  k 2  (f> 1  —  k  sin  (<f)  -j-  ^ ' ) 

H,  —  (f)  —  k2  (f>'  T  k  sin  (4>  —  (/>') 

,  «  sin  ((f)  —  (f)') 

p 

sin  (f) 


8c/>  — 


BET, 


2  k  tg  (f) '  sin  ((f)--  (f> '  ) 


/  was  obtained  by  subtracting  the  ordinate  of  Argelander’s 
curve  for  the  selected  instants  from  the  mean  of  the  maximum 
ordinates,  calling  this  difference  A  G,  and  computing  J  from  the 
equation  log  /=  0.051  A  G ,  which  is  readily  derived  from  Pog- 
sjii’s  scale,  together  with  the  value  of  Argelander’s  grade.  The 


SYSTEM  OF  (3  LYRAE 


3 


values  of  MIy  Min  Hf,  HIV  are  computed  directly  from  values  of 
J  obtained  from  the  above  curve,  and  then  an  approximate  value 
of  <£,  interpolated  from  tables  computed  from  the  M's  and  H’s 
with  (f)  as  an  argument,  are  then  corrected  by  the  above  differen¬ 
tial  formulae.  Thus  it  is  possible  by  these  formulae  to  compute  a 
light  curve  from  known  elements,  and  also  to  solve  the  converse 
problem.  To  compute  these  tables  the  values  of  k,  were  required. 
k  being  unknown,  an  approximation  of  its  value  was  obtained 
thus  :  calling  bm  the  brightness  of  a  star  of  magnitude  m ,  and 
bM  that  of  a  star  of  magnitude  M,  by  Pogson’s  scale 

log  =  log  /  0.4  A  M, 

where  £s,M  i s  the  difference  of  the  brightness  in  stellar  magni¬ 
tudes.  From  this  we  have 


Brightness  at  Min.  II 
Brightness  at  Min.  I 


1-8536 


Brightness  at  Max. 
Brightness  at  Min.  I 

The  first  of  these  gives 


(<0 

and  the  second 

w 


I  +  K  2  A —  a  k  2  A 

I  — a  K2  -j-  K2  A 


I  — j—  a*2  +  K2  A 
I  —  a  k2  -j-  k2  A 


m, 


where  a  denotes  the  portion  of  the  disks  common  to  both  bodies 
at  the  middle  of  the  eclipses. 

By  a  few  simple  transformations  of  these  equations,  it  may 
be  readily  seen  that 

q  >  1.0205,  *•  e-  the  disks  must  be  flattened,  and  0.8323  1.5241. 

The  values  of  /c  selected,  were  then 

0.8323,  0.9049,  1. 0000,  1.2883  and  *-5241 

and  a  table  such  as  was  mentioned  above  was  computed  with  the 
argument  <£  or  <£'  according  as  k<  i,  or  k>  i. 


G.  W.  MYERS 


14 


The  values  of  ry  for  various  points  before  and  after  the  min¬ 
ima,  with  a  correct  hypothesis  for  re  and  i,  the  inclination,  must 
all  be  approximately  equal,  since  they  are  computed  on  the 
hypothesis  of  a  circular  orbit,  k-  1.524 1  gave  a  fair  approach 
to  an  agreement  in  the  various  values,  while  the  other  values  of 
/c  gave  widely  discordant  results  for  r.  For  some  reasons  it  was 
more  convenient  to  have  k<  i,  and  hence,  the  larger  radius  was 
assumed  unity  and  the  necessary  alterations  in  the  formulae  were 
made  to  suit  this  hypothesis.  The  only  possible  assumption 

which  could  be  made  for  i  was  shown  to  be  -,  since  this  gave 

the  various  values  of  r  more  nearly  equal  than  any  other  assump¬ 
tion  for  it.  A  was  found  to  be  0.3353  from  ( d )  and  (^). 
Recapitulating  then;  the  following  values  were  taken  as  first 
approximations  : 


e  -  --  o,  i=  - ,  r  1 .8344,  k  0.6  q6 1 ,  A  ~  0.3353  ar|d  <7=1.2. 

2 

By  differentiating  the  above  formulae  and  combining  the 
results,  the  following  differential  equations  for  correcting  the 
approximate  circular  elements  were  derived  : 

(I  )*(\  +  *)JJ,=  2KKld*+  J"Sin*'  dr  +  r*  C0S,gsin*'  d  (.■' ») 

r  o 


d' 


•0» 


where  Kf=  A  Jt  f2  cos2  — /,) — /<£ —  Cpf2  cos2  (3  sin 


and 


A  —  Trq\,  J3  -  tt  ^  1  +  ^  ^  k2 — 


C- 


2?3  ‘ 


m 


q  must  not  be  included  in  this  equation  since  it  depends  on  k 
by  equation  (e).  The  equation  applies  only  during  Min.  I.  For 
Min.  II, 

/  t  T  \  tt(A+k2)  7r  ^  7  ,  2  p  sin  <f>'  ,  r2  cos2  /?  sin  <j>'  , 

(II)  — ^ - dJ//=2K  KndK  -| - r -  dr-\ - d  (i  )2 

A  r  p 

K„=  A  J"  f-x  —  P  +  tt (/—//,)—  Cp/2  cos2  p  sin  . 


SYSTEM  OF  p  LYRAE 


15 


The  coefficients  of  (I)  and  (II)  were  computed  for  14  points 
selected  symmetrically  along  the  curve,  with  the  approximate 
values  given  above,  dj,  and  dj n  were  obtained  by  comparing 
corresponding  ordinates  of  Argelander’s  curve  with  those  of  a 
light  curve  computed  from  the  preceding  approximate  values. 
14  observation  equations  lead  to  the  following  values: 

dK  —  -f  0.3586,  dr—  -j-  0.0999,  and  d(i'z)  =  — 0.0434 

i’  being  here  imaginary,  but  numerically  small,  it  was  called  0, 
and  dK  being  quite  large,  dq  was  introduced  in  its  stead,  since 
small  changes  in  the  fundamental  data  do  not  affect  q  so  greatly 
as  r,  and  the  equations  again  solved,  gave  the  corrections, 
dq  ~=  —  0.0007,  and  dr= — 0.0889,  and  the  corrected  values 
were  then  <71. 1993  and  r  1.8955,  and  from  ( d )  and  ( e ), 
k  0.7580  and  X  — 0.4023.  The  probable  error  was,  of  course, 
somewhat  increased  by  dropping  V ,  being  in  the  latter  case 
±0.1  of  a  “brightness.”  These  values  were  regarded  as  the 
most  probable  on  the  assumption  of  a  circular  orbit. 

By  means  of  these  values  and  differential  equations,  which 
were  derived  for  correcting  a  circular  into  an  elliptical  orbit  of 
small  eccentricity,  a  set  of  48  observation  equations  was  obtained, 
for  as  many  points  of  the  light  curve,  and  from  them  the  follow¬ 
ing  elliptical  elements  were  obtained  : 


T  1855  Jan  13d  6\35 

P 

12.91  d 

0=i-937 

=  27°. 887 

i= 90  0 

K  - 

=  0.7528 

rj  940  53'  from  node 

A 

o-399 

0=-o°  (assumed) 

q- 

=  1.203 

t  =  6d  i5h-4  after  Min.  I. 

Both  e  and  1  were  included  in  the  equation  and  i'  was  again 
imaginary  but  very  small. 

Using  the  equations  derived  for  computing  the  light  curve 
for  elliptical  motion,  the  results  of  computation  are  comprised 
in  the  columns  of  the  table  given  below.  The  columns  headed 
A  andyB  contain  the  computed  and  observed  ordinates  respec¬ 
tively,  and  those  headed  A/B_R,  the  corresponding  residuals 


G.  W.  MYERS 


16 

from  Argelander’s  curve.  The  computed  curve  is  shown  in 
(Fig.  irt)  drawn  in  a  dotted  line  beside  Argelander’s  curve 
drawn  in  a  full. line.  The  agreement  is  quite  close. 

TABLE  OF  COMPUTED  AND  OBSERVED  BRIGHTNESS. 


Minimum  I 

Minimum  II 

•A 

5/b-r 

S-'B-R 

—72 

0.9963 

O.9956 

'  +0.0007 

O.9918 

O.9970 

— 0.0052 

—66 

0.9870 

O.9852 

+0.0018 

O.9836 

0-9945 

—0.0109 

— 60 

0.9683 

0.9701 

— 0.0018 

O.9696 

0.9833 

—0.0137 

—54 

0.9524 

0-9513 

+0.001  I 

O.949O 

O.9672 

— 0.0182 

-48 

0.9147 

O.9270 

— 0.0123 

O.9258 

O.9482 

— 0.0224 

—42 

0.8732 

O.8849 

- 0.0 1 17 

0.8995 

O.9217 

— 0.0222 

— 36 

0.8209 

0.8268 

— 0.0059 

0.8680 

0.8886 

— 0.0206 

—3° 

0.7296 

0.7525 

— 0.0229 

O.8306 

0.8494 

— 0.0188 

—24 

0.5836 

O.6CI9 

— 0.0183 

O.7836 

0.8048 

— 0.0212 

—  18 

0-4336 

0.4993 

— 0.0657 

O.7246 

0.7558 

— 0.0312 

— 12 

0.3661 

0.4275 

— 0.0614 

O.6674 

0.7044 

—0.0370 

-  6 

0.3484 

O.3487 

— 0.0003 

O.6433 

0.6542 

— 0.0109 

0 

0-3433 

0-3433 

+0 

O.6365 

0.6368 

— 0.0003 

+  6 

0.3499 

O.3488 

+0.001 1 

O.6472 

0.6372 

+  0.0100 

+12 

0.3988 

0.4275 

— 0.0287 

O.6762 

0.6689 

+0.0073 

+ 18 

0.5306 

0.5591 

— 0.0285 

0.7340 

0.7203 

+O.OI37 

+24 

0.6572 

O.6624 

+0.0052 

O.7978 

0.7716 

+0.0262 

+30 

0.7644 

O.7528 

+0.01 16 

O.8498 

0.8289 

+O.O2O9 

+36 

0.8416 

O.8266 

+0.0150 

O.893I 

0.8622 

+O.O3O9 

+42 

0.8857 

O.8845 

+0.0012 

0.9255 

0.8997 

+0.0258 

+48 

0.9234 

O.9268 

—0.0034 

0.9524 

0.9307 

+0.0217 

+54 

0-9537 

O.9508 

+0.0029 

O.9627 

0-9545 

+  0.0082 

+60 

0.9732 

O.9694 

+0.0038 

O.9881 

0.9732 

+O.OI49 

+  66 

0.9870 

O.9847 

+0.0023 

0.9977 

0.9877 

+0.0100 

+72 

0.9940 

0.9952 

— 0.0012 

I.OO49 

0.9970 

+0.0079 

With  the  help  of  an  eccentricity,  therefore,  the  residuals  are 
somewhat  reduced,  and,  considering  the  errors  necessarily  attach¬ 
ing  to  photometric  estimates,  the  curve  of  Argelander  may  be 
regarded  as  sufficiently  well  represented. 

The  eccentricity  results  again  almost  the  same  as  before,  so 
that  at  the  epoch  1855,  the  eccentricity  did  not  differ  materially 
from  0.02. 

Since  the  periastron  lies  near  Min  II,  the  systematic  devia¬ 
tion  of  the  computed  from  Argelander’s  curve,  the  former  lying 
above  the  latter,  before,  and  below  it,  after  Min  II,  it  is  highly 
probable,  that  while  the  secondary  rounds  periastron,  very  con- 


SYSTEM  OF  /3  LYRAE 


'7 

siderable  augmentation  of  brightness  due  to  deformations  of  the 
disks,  to  internal  friction,  etc.,  occurs.  Because  of  inertia,  these 
effects  could  not  immediately  show  themselves,  so  that  the  real 
curve  would  lie  below  the  computed  mean  before  Min  II  and 
above  it  after  Min  II,  as  is  represented  in  the  figure. 

Feeling  somewhat  suspicious  that  any  one  of  several  sets  of 
elements  lying  near  those  just  given  might  represent  observa¬ 
tions  equally  well,  it  seemed  worth  while  to  test  whether  this 
same  set  of  elements  would  result,  if  one  of  the  circular  elements 
upon  which  the  elliptical  ones  were  based,  were  given  an  arbi¬ 
trary  change  and  then,  by  repeated  applications  of  the  method 
of  least  squares,  the  elements  were  again  corrected  by  the  obser¬ 
vations.  An  arbitrary  change  of  — 0.01  was  given  to  k.  and  of 
—  0.07  to  r  and  after  four  adjustments  giving  ever  smaller  proba¬ 
ble  errors,  the  former  values  were  again  obtained. 

It  may  therefore  be  inferred  that  the  above  elements  are  the 
most  probable.  An  interesting  fact  arising  during  the  latter 
process  of  adjustment,  was  that  one  set  of  elements  giving  a 
probable  error  of  nearly  the  same  magnitude  as  the  final  set, 
gave  the  distance  between  the  centers  =fe§  1.80  and  the  sum  of  the 
radii  equal  to  1.82,  i.  e.}  the  components  are  not  yet  separated. 
This  fact  in  connection  with  the  low  mean  density  of  the  system 
points  to  the  nebulous  condition  of  the  star.  The  indications  are 
then,  either  that  the  companions  are  not  yet  separate,  but  in  the 
act  of  separation,  or  that  if  separate,  their  separation  has  taken 
place  comparatively  recently.  In  either  case  we  seem  to  have 
here  the  first  concrete  example  of  a  world  in  the  act  of  being 
born. 


SPECTROSCOPIC  CONSIDERATIONS. 

Treating  by  the  method  of  Rambaut,  published  in  Mon.  Not. 
51,  316,  the  observations  of  Belopolsky  made  between  Sept.  23 
and  Nov.  26,  1892,  and  published  in  Melanges  math,  et  astrono- 
miques ,  t.  VII,  l.  3,  I  find, 

V  =  —  0.8  kilometers  per  second 
T—  26.93  September  1892 


G.  IV.  MYERS 


I  8 


e  =  0.108 

o>  — 790  17'  from  node 
a  sin  i  —  15836000  kilometers 

P  =  i2d.9i  from  light  period. 

Lockyer  observes  the  relative  displacements  of  three  lines 
and  finds  velocities  as  follows  : 

Hy  —  155.0  miles 

178=  154-0  miles 

^4025=  i58-°  mdes. 

The  mean  of  these  values  is  155.7  miles  and  from  a  private 
letter  of  Professor  Lockyer,  1  find  it  belongs  to  the  epoch, 
August  24.46,  1893.  Correcting  for  the  motion  of  the  Earth  and 
reducing  to  kilometers,  there  results  for  the  relative  velocity  in 
the  line  of  sight  259.8  kilometers. 

Belopolsky’s  measures  give  for  the  diameter  of  the  absolute 
orbit  of  one  of  the  components  31672000  kilometers.  From 
the  absolute  orbit  furnished  by  Belopolsky’s  observations,  the 
velocity  for  the  above  date  was  found  to  be  of  the  above 

relative  velocity.  The  semi-major  axis  of  the  relative  orbit  is 
then  A  =  3. 168/2 X  3 1672000  —  501 75000  kilometers  (calling 

7T 

i  =~  -  )  and  for  the  ratio  of  the  masses 

2 

a  '  A  =  m  M),  or  m  / M  = - —  • 

Assuming  that  the  F-line  observed  by  Belopolsky  was  pro¬ 
duced  by  the  smaller  component,  we  have 

A  /a  =  (M m )  / m  =  1  -|-  8 '  k3, 

(S'  — ratio  of  densities  of  components), 
and  assuming  his  line  to  be  produced  by  the  larger,  there  results, 
A  /a  =  (M - f-  m)  / m  1  -j-  1  S'k3. 

Using  the  values  of  a / A  and  k  given  above,  we  find 
8'=  5.083,  or  1. 081. 

This  furnishes  a  means  of  deciding  which  of  the  components 
produced  Belopolsky’s  F-line.  Since  it  is  quite  improbable  that 
one  of  the  two  bodies,  so  near  to  each  other  as  is  the  case  here, 
should  have  a  density  5.083  times  that  of  the  other,  the  latter 


SYSTEM  OF  ft  LYRAE 


19 


value  is  assumed  to  be  the  correct  one,  and  hence,  that  Belopol¬ 
sky’s  F-line  was  due  to  the  larger  component. 

Using  now  the  well  known  relation 

M+m  =  A*  /F* 

and  substituting  the  values  of  A  in  astronomical  units  and  of 
P  in  years,  there  result, 

M- f-  m  —  30.56  solar  masses 
and  since  m/ M=  1  /  2. 168, 

M=  20.91  solar  masses 
and  m  —  9.65  solar  masses. 

Calling  5  the  solar  mass,  H  the  solar  radius,  R'  and  R,  the 
semi-major  and  semi-minor  axes  of  the  minor  ellipsoid,  the 
density  of  the  larger  companion  in  terms  of  the  solar  density 
and  S2  of  the  smaller, 

Sj  =  (M/S)  q*  (H  /R')z,  (q~R'  R  as  before) 

and  substituting  former  values, 

=  0.00058. 

But 

82  =  1.08  Sj  (8'  =  1.08) 

,  hence 

82  —  0.00063. 

The  mean  density  of  the  system  is  then  somewhat  less  than 
the  density  of  air,  i.  e .,  comparable  to  nebular  density.  It 
appears  then  that  (3  Lyrae  furnishes  us  a  concrete  illustration  of 
the  actual  existence  in  space  of  a  Poincare  figure  of  equilibrium. 

Using  the  chief  epochs  of  Lindemann’s  curve  constructed 
from  Plassmann’s  observations,  the  following  equations  result : 

cos  (Mx  +  49°  57')  =  — 0.0563/^  =  —  3*204  (with  Argelander’s  e) 
cos  (Mt  4-  930  52')  =  —  0.0338/V  =  —  1.920 
cos  ( M1  -)-  136°  2 3 '.5)  =  —  0.0177 1  / e  —  —  1. 010. 

The  impossibility  of  these  relations,  on  the  hypothesis  that  e 
is  not  greater  than  it  was  in  1855,  requires  us  to  infer  that  since 
Argelander’s  time  the  eccentricity  of  the  system  has  grown 
larger.  Comparing  the  spectroscopic  with  the  photometric 


20 


G.  IV.  MYERS 


observations,  it  is  also  seen  that  a  large  motion  of  the  line  of 
apsides  has  occurred  since  1855,  though  its  precise  amount 
can  scarcely  be  ascertained  with  any  considerable  degree  of 
certainty. 

The  following  results,  therefore,  seem  to  be  quite  clearly 
indicated  by  the  preceding  discussion  : 

1.  The  photometric  estimates  of  /3  Lyrae’s  variability  may  be 
explained  easily  within  the  limits  of  the  errors  of  these  esti¬ 
mates  by  the  aid  of  the  satellite  theory. 

2.  The  bodies  may  be  regarded  as  similar  ellipsoids  of  revo¬ 
lution. 

3.  The  orbit  of  the  secondary  body  is  nearly  circular  and  its 
plane  passes  almost  exactly  through  the  Sun. 

4.  The  common  flattening  of  the  ellipsoids  differs  but  little 
from  0.17  and,  aside  from  librations,  the  periods  of  rotation  and 
revolution  are  equal. 

5.  The  larger  body  is  about  0.4  as  bright  as  the  smaller. 

6.  The  distance  of  centers  is  extremely  small,  about  \  l/%  of 
the  semi-major  axis  of  the  larger  ellipsoid. 

7.  From  Lindemann’s  chief  epochs  for  1892  the  orbital 
eccentricity  of  the  system  must  have  increased  from  1855. 
Belopolsky’s  spectroscopic  observations  indicate  the  same. 

8.  The  motion  of  the  center  of  gravity  of  the  system  with 
respect  to  the  Sun  is  very  small. 

9.  The  semi-major  axis  of  the  orbit  of  the  companion  is 
about  50,000,000  kilos. 

10.  The  mass  of  the  larger  body  is  21  times,  and  of  the 
smaller  9.5  times  the  solar  mass. 

1 1.  The  densities  of  the  companions  are  nearly  the  same. 

12.  The  mean  density  of  the  system  is  comparable  with 
atmospheric  density,  or  the  “system”  (for  such,  I  think,  it 
must  now  be  called),  is  in  a  nebulous  condition. 

13.  In  conclusion,  it  may  be  said  that  the  spectroscopic 
and  photometric  observations,  which  were  available  to  me  for 
the  foregoing  discussion,  so  far  from  being  widely  discordant, 
as  some  have  thought,  agree  with  each  other  remarkably  closely. 


SYSTEM  OF  LYRAE 


21 


The  strong  absorption  lines  in  the  spectrum  of  this  star  point 
to  the  presence  of  a  powerfully  absorbing  atmospheric  envelope 
about  the  nuclei  of  the  masses.  From  the  dynamical  theory  of 
gases  we  know  that  such  an  atmospheric  layer  would  arrange 
itself  about  the  combined  mass  of  the  system  so  that  portions 
of  equal  density  would  be  in  equipotential  surfaces.  These 
surfaces  would,  in  the  immediate  vicinity  of  the  surfaces  of  the 
masses,  conform  somewhat  closely  to  the  surface  of  the  bodies, 
but  they  would  rapidly  lose  the  abrupt  curvatures  at  the  surface 
and  become  more  and  more  nearly  spherical.  A  rough  attempt 
to  represent  this  is  shown  in  the  subjoined  figure. 


The  equipotential  surfaces,  shown  by  dotted  lines,  would 
dispose  themselves  svmmetrically  about  the  center  of  gravity 
of  the  system  as  a  center,  and  if  the  density  of  the  larger  com¬ 
panion  were  very  materially  greater  than  that  of  the  smaller, 
the  center  of  gravity  would  lie  far  within  the  larger  compan¬ 
ion.  It  might  even  happen  that  the  center  of  gravity  of  the 
two  bodies  should  lie  beyond  the  geometrical  center  of  the 
larger  ellipsoid.  Where  it  would  lie  would  depend  wholly  upon 
the  density  of  the  various  parts  of  the  mass  of  the  two  bodies. 
It  is  then  readily  seen  that  the  atmosphere  might  be,  and  prob¬ 
ably  is,  so  arranged  as  to  permit  the  remote  end  of  the  smaller 


22 


G.  W.  MYERS 


to  shine  through  a  shallow  layer  of  it  and  thereby  to  permit 
the  smaller  to  appear  brighter,  even  though  it  might  be  intrin¬ 
sically  darker  than  the  larger  body.  No  violence  is  done  to 
theory,  at  all  events,  by  such  an  assumption.  This  distribution 
of  the  atmosphere  would  also  explain  the  absorption  bands, 
which  are  seen  in  the  spectrum  of  this  star  more  distinctly  at 
Min  I  than  at  Min  II.  The  continuous  spectrum,  which  would 
be  produced  most  distinctly  by  the  smaller  body,  must  at  the 
same  time  appear  fainter.  This  accords  with  spectroscopic 
observations  also. 

Although  some  of  the  ideas  given  above  may  seem  a  little 
venturesome,  let  it  be  remembered  that  the  peculiar  character 
of  the  observations  of  this  star  leads  one  to  expect  an  explana¬ 
tion  of  a  somewhat,  unusual  nature. 

That  the  ellipsoids  are  similar  is,  of  course,  an  arbitrary 
assumption. 

In  conclusion,  let  it  be  observed  that  an  attempt  at  a  formal 
representation  of  the  condition  of  things  prevailing  in  the  sys¬ 
tem  of  ft  Lyrae,  leads  to  the  assumption  of  a  single  body  (such 
as  Poincare’s  or  Darwin’s  figures  of  equilibrium) .  The  above 
has,  of  course,  only  a  formal  significance,  but  on  account  of  the 
poverty  of  observational  material  at  my  disposal  an  attempt  to 
push  the  discussion  farther  on  a  mathematical  basis  could  not 
have  proved  profitable.  It  is  believed,  however,  that  the  discus¬ 
sion  may  help  us  to  orient  our  views  with  regard  to  this  wonder¬ 
fully  interesting  star.  Fig.  i b  represents  the  most  probable 
relative  dimensions  of  the  bodies  and  orbit  of  the  system,  as 
based  on  Argelander’s  photometric  estimates  up  to  1859.  Pro¬ 
fessor  Pickering  has  kindly  offered  to  place  all  the  earlier  esti¬ 
mates  of  /3  Lyrae’s  brightness  at  the  writer’s  disposal,  and  it  is 
the  latter’s  intention  to  make  a  full  investigation  and  discussion 
of  them  at  the  earliest  possible  date. 


